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Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 381571, 9 pages
http://dx.doi.org/10.1155/2011/381571
Research Article

Delay-Dependent Stability Analysis and Synthesis for Uncertain Impulsive Switched System with Mixed Delays

Naval Aeronautical Engineering Institute Qingdao Branch, Qingdao 266041, China

Received 27 November 2010; Accepted 15 March 2011

Academic Editor: Guang Zhang

Copyright © 2011 Binbin Du and Xiaojie Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper studies the asymptotic stability problem for a class of uncertain impulsive switched systems with discrete and distributed delays. Based on Lyapunov functional theory, delay-dependent sufficient LMI conditions are established for the asymptotic stability of the considered systems. Moreover, an appropriate feedback controller is constructed for stabilizing the corresponding closed-loop system. The results are illustrated to be efficient through an example.

1. Introduction

A switched system is a type of hybrid system which is a combination of discrete and continuous dynamical systems. These systems arise as models for phenomena which cannot be described by exclusively continuous or exclusively discrete processes. Recently, on the basis of Lyapunov functions and other analysis tools, the stability and stabilization for switched systems have been investigated and many variable results have been obtained; see [14]. In general, the switched systems which have been widely studied in the literature can be classified into two groups: continuous switched systems and discrete switched systems. However, both of these classes do not cover some useful switched systems existing in the real world displaying a certain kind of dynamics with impulse effect at the switching points, that is, the states jump. Studies on the dynamic systems with impulsive effect and switching have arisen in various fields of science and engineering in recent years; see [511]. These systems are called impulsive switched systems, which are useful to model those physical phenomena that exhibit abrupt changes at certain time points due to impulsive inputs or switching. For these systems, there is an increasing interest among the control community in terms of stability analysis and the design of stabilizing feedback controller so as to achieve a required stability performance. For example, in [6], the authors studied a class of uncertain impulsive switched systems with delay input; in [7], the author studied a class of impulsive switched systems; by constructing appropriate Lyapunov-Krasovskii functions and using LMI approach, some asymptotic stability criteria were obtained and some appropriate feedback controllers were constructed. To the best of the authors’ knowledge, most of the papers have studied the delay-independent stability criteria, and few delay-dependent results have been reported in the literature concerning the problem of robust stability for the impulsive switched systems. This motivates our research.

On the other hand, time delays and uncertainties happen frequently in various engineering, biological, and economical systems, and they many result in instability. Many stability criteria have been derived for continuous dynamical systems with time delays or uncertainties; see [1214]. However, such fewer results have been reported for stability analysis and control of impulsive switched systems with distributed time delays.

In this paper, the problem of delay-dependent stability analysis and synthesis for impulsive switched system with discrete and distributed delays is studied. The uncertainties under consideration are norm bounded. Based on Lyapunov functional approach and linear matrix inequality technology, some new delay-dependent stability and stabilization conditions are derived. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.

Notations. Throughout the paper, 𝐴𝑇 stands for matrix transpose of the matrix 𝐴. 𝑅𝑛 is the 𝑛-dimensional Euclidean space. 𝑅𝑛×𝑚 is the set of all 𝑛×𝑚-dimensional matrices. 𝐼 denotes the identity matrix of appropriate dimensions. 𝑃>0(𝑃<0) means that 𝑃 is a symmetric positive definite (negative definite) matrix. represents the elements below the main diagonal of a symmetric matrix.

2. Problem Formulation and Preliminaries

Consider the following impulsive switched system with mixed delays:̇𝑥(𝑡)=𝐴𝑖𝑘𝑥(𝑡)+𝐵𝑖𝑘𝑥𝑡1+𝐶𝑖𝑘𝑡𝑡2𝑥(𝑠)d𝑠+𝐷𝑖𝑘𝑢(𝑡),𝑡𝑡𝑘,Δ𝑥(𝑡)=𝐺𝑘𝑥(𝑡),𝑡=𝑡𝑘,𝑥(𝑡)=𝜑(𝑡),𝑡0,(2.1) where 𝑥(𝑡)𝑛 and 𝑢(𝑡)𝑝, 𝑛,𝑝, are the state vector and the control input, respectively. 1>0, 2>0 are time delays, =max{1,2}. Δ𝑥(𝑡)=𝑥(𝑡+)𝑥(𝑡), where 𝑥(𝑡+)=lim𝑣0+𝑥(𝑡+𝑣), 𝑥(𝑡)=lim𝑣0+𝑥(𝑡𝑣). 𝑥(𝑡𝑘)=𝑥(𝑡𝑘), which means that the solution of the system (2.1) is left continuous at the impulsive switched time point 𝑡𝑘 which satisfies 𝑡0<𝑡1<<𝑡𝑘<, lim𝑘+𝑡𝑘=+. 𝑖𝑘{1,2,,𝑚}, 𝑘,𝑚, is a discrete state variable. {𝑖𝑘,𝑡𝑘} represents a switching rule of the system (2.1), that is, at 𝑡𝑘 time point, the system switches to the 𝑖𝑘 subsystem from the 𝑖𝑘1 subsystem:𝐴𝑖𝑘=𝐴𝑖𝑘+Δ𝐴𝑖𝑘(𝑡),𝐵𝑖𝑘=𝐵𝑖𝑘+Δ𝐵𝑖𝑘(𝑡),(2.2)𝐴𝑖𝑘,𝐵𝑖𝑘,𝐶𝑖𝑘,𝐺𝑘𝑛×𝑛, 𝐷𝑖𝑘𝑛×𝑝 are known constant real matrices. Δ𝐴𝑖𝑘() and Δ𝐵𝑖𝑘() are unknown real norm-bounded matrix functions which represent time-varying parameter uncertainties, which are of the following form:Δ𝐴𝑖𝑘(𝑡)Δ𝐵𝑖𝑘(𝑡)=𝐸𝑖𝑘𝐹𝑖𝑘𝐻(𝑡)𝑖𝑘𝐽𝑖𝑘,(2.3) where 𝐸𝑖𝑘, 𝐻𝑖𝑘, and 𝐽𝑖𝑘 are known constant real matrices of appropriate dimensions and 𝐹𝑖𝑘(𝑡) is an unknown real time-varying matrix satisfying 𝐹𝑇𝑖𝑘(𝑡)𝐹𝑖𝑘(𝑡)𝐼.

Lemma 2.1. Let 𝐷, 𝐸, and 𝐹 be matrices with appropriate dimensions. Suppose that 𝐹𝑇𝐹𝐼; then, for any real scale 𝜆>0, one has that 𝐷𝐹𝐸+𝐸𝑇𝐹𝑇𝐷𝑇𝜆𝐷𝐷𝑇+𝜆1𝐸𝑇𝐸.(2.4)

Lemma 2.2. For any constant matrix 𝑀𝑛×𝑛, 𝑀>0, and a scalar 𝛾>0, vector function 𝑤is such that the integrations concerned are well defined; then 𝛾0𝑤(𝑠)ds𝑇𝑀𝛾0𝑤(𝑠)ds𝛾𝛾0𝑤𝑇(𝑠)𝑀𝑤(𝑠)ds.(2.5)

3. Stability Analysis

Theorem 3.1. Suppose that there exist symmetric positive definite matrices 𝑃𝑖𝑘, 𝑄𝑖𝑘, 𝑇𝑖𝑘, and 𝑅𝑖𝑘 and some positive scalars 𝜀1,𝜀2 such that for 𝑖𝑘=1,2,,𝑚 the following LMIs hold: (a)𝑄𝑖𝑘00𝑄𝑖𝑘01𝑇𝑖𝑘01𝑇𝑖𝑘02𝑅𝑖𝑘2𝑅𝑖𝑘0𝑍𝑖𝑘𝑃𝑖𝑘𝐸𝑖𝑘𝜀1+𝜀21𝐼<0,(3.1) where 𝑍𝑖𝑘=𝑃𝑖𝑘𝐴𝑖𝑘+𝐴𝑇𝑖𝑘𝑃𝑖𝑘+𝜀11𝐻𝑇𝑖𝑘𝐻𝑖𝑘+𝐼,(b)𝐼𝑃𝑖𝑘𝐵𝑖𝑘0𝑃𝑖𝑘𝐶𝑖𝑘𝑄𝑖𝑘+𝜀11𝐽𝑇𝑖𝑘𝐽𝑖𝑘1001𝑇𝑖𝑘012𝑅𝑖𝑘<0,(3.2)(c)𝑃𝑖𝑘1𝐼+𝐺𝑘𝑇𝑃𝑖𝑘𝑃𝑖𝑘>0.(3.3) Then the trivial solution of the impulsive switched system (2.1) with 𝑢(𝑡)=0 is robustly asymptotically stable.

Proof. When 𝑡(𝑡𝑘,𝑡𝑘+1], consider the Lyapunov-Krasovskii function candidate 𝑉(𝑥(𝑡))=𝑥𝑇(𝑡)𝑃𝑖𝑘𝑥(𝑡)+𝑡𝑡1𝑥𝑇(𝑠)𝑄𝑖𝑘+𝑥(𝑠)d𝑠01𝑡𝑡+𝑠𝑥𝑇(𝑢)𝑇𝑖𝑘𝑥(𝑢)d𝑢d𝑠+02𝑡𝑡+𝑠𝑥𝑇(𝑢)𝑅𝑖𝑘𝑥(𝑢)d𝑢d𝑠.(3.4) Taking the right upper derivative of 𝑉(𝑥(𝑡)) along the solution of the impulsive switched system (2.1), we have that 𝐷+𝑉(𝑥(𝑡))𝑥𝑇(𝑡)2𝑃𝑖𝑘𝐴𝑖𝑘+𝐸𝑖𝑘𝐹𝑖𝑘𝐻𝑖𝑘+𝑄𝑖𝑘+1𝑇𝑖𝑘+2𝑅𝑖𝑘𝑥(𝑡)𝑥𝑇𝑡1𝑄𝑖𝑘𝑥𝑡1+2𝑥𝑇(𝑡)𝑃𝑖𝑘𝐵𝑖𝑘+𝐸𝑖𝑘𝐹𝑖𝑘𝐽𝑖𝑘𝑥𝑡1𝑡𝑡1𝑥𝑇1(𝑠)d𝑠1𝑇𝑖𝑘𝑡𝑡1𝑥(𝑠)d𝑠𝑡𝑡2𝑥𝑇1(𝑠)d𝑠2𝑅𝑖𝑘𝑡𝑡2𝑥(𝑠)d𝑠+2𝑥𝑇(𝑡)𝑃𝑖𝑘𝐶𝑖𝑘𝑡𝑡2𝑥(𝑠)d𝑠.(3.5) Define 𝑥𝜉(𝑡)=𝑥(𝑡)𝑡1𝑡𝑡1𝑥(𝑠)d𝑠𝑡𝑡2𝑥(𝑠)d𝑠,Φ𝑖𝑘=Φ11Φ120𝑃𝑖𝑘𝐶𝑖𝑘𝑄𝑖𝑘1001𝑇𝑖𝑘012𝑅𝑖𝑘,(3.6)Φ11=𝑃𝑖𝑘𝐴𝑖𝑘+𝐴𝑇𝑖𝑘𝑃𝑖𝑘+𝑄𝑖𝑘+1𝑇𝑖𝑘+2𝑅𝑖𝑘+𝑃𝑖𝑘𝐸𝑖𝑘𝐹𝑖𝑘𝐻𝑖𝑘+𝐻𝑇𝑖𝑘𝐹𝑇𝑖𝑘𝐸𝑇𝑖𝑘𝑃𝑖𝑘, and Φ12=𝑃𝑖𝑘𝐵𝑖𝑘+𝑃𝑖𝑘𝐸𝑖𝑘𝐹𝑖𝑘𝐽𝑖𝑘; then we have that 𝐷+𝑉(𝑥(𝑡))𝜉𝑇(𝑡)Φ𝑖𝑘𝜉(𝑡).(3.7) Let Ψ𝑖𝑘=𝑃𝑖𝑘𝐴𝑖𝑘+𝐴𝑇𝑖𝑘𝑃𝑖𝑘+𝑄𝑖𝑘+1𝑇𝑖𝑘+2𝑅𝑖𝑘𝑃𝑖𝑘𝐵𝑖𝑘0𝑃𝑖𝑘𝐶𝑖𝑘𝑄𝑖𝑘1001𝑇𝑖𝑘012𝑅𝑖𝑘.(3.8) Then by Lemma 2.1 and Schur complement, we have that Φ𝑖𝑘=Ψ𝑖𝑘+𝑃𝑖𝑘𝐸𝑖𝑘𝐹𝑖𝑘𝐻𝑖𝑘+𝐻𝑇𝑖𝑘𝐹𝑇𝑖𝑘𝐸𝑇𝑖𝑘𝑃𝑖𝑘𝑃𝑖𝑘𝐸𝑖𝑘𝐹𝑖𝑘𝐽𝑖𝑘,00000000Ψ𝑖𝑘+𝜀1+𝜀2𝑃𝑖𝑘𝐸𝑖𝑘𝐸𝑇𝑖𝑘𝑃𝑖𝑘+𝜀11𝐻𝑇𝑖𝑘𝐻𝑖𝑘000𝜀21𝐽𝑇𝑖𝑘𝐽𝑖𝑘,=00000𝐼𝑃𝑖𝑘𝐵𝑖𝑘0𝑃𝑖𝑘𝐶𝑖𝑘𝑄𝑖𝑘+𝜀21𝐽𝑇𝑖𝑘𝐽𝑖𝑘1001𝑇𝑖𝑘012𝑅𝑖𝑘+𝑍𝑖𝑘,000000000(3.9) where 𝑍𝑖𝑘=𝑃𝑖𝑘𝐴𝑖𝑘+𝐴𝑇𝑖𝑘𝑃𝑖𝑘+𝑄𝑖𝑘+1𝑇𝑖𝑘+2𝑅𝑖𝑘+(𝜀1+𝜀2)𝑃𝑖𝑘𝐸𝑖𝑘𝐸𝑇𝑖𝑘𝑃𝑖𝑘+𝜀11𝐻𝑇𝑖𝑘𝐻𝑖𝑘+𝐼.
The stability condition 𝐷+𝑉(𝑥(𝑡))<0 can be obtained if the following inequalities hold: 𝐼𝑃𝑖𝑘𝐵𝑖𝑘0𝑃𝑖𝑘𝐶𝑖𝑘𝑄𝑖𝑘+𝜀21𝐽𝑇𝑖𝑘𝐽𝑖𝑘1001𝑇𝑖𝑘012𝑅𝑖𝑘<0,𝑍𝑖𝑘<0.(3.10) By condition (b) of the theorem, the former inequality is satisfied. 𝑍𝑖𝑘<0 will hold if the following condition is satisfied: diag𝐼,𝐼,𝐼,𝑍𝑖𝑘,𝐼<0.(3.11)
Define 𝑊𝑖𝑘=𝑄𝑖1/2𝑘0000011/2𝑇𝑖1/2𝑘0000021/2𝑅𝑖1/2𝑘00𝑄𝑖1/2𝑘11/2𝑇𝑖1/2𝑘21/2𝑅𝑖1/2𝑘𝜀𝐼1+𝜀21/2𝑃𝑖𝑘𝐸𝑖𝑘𝜀00001+𝜀21/2𝐼.(3.12) Then by left multiplying and right multiplying (3.11) by 𝑊𝑖𝑘 and 𝑊𝑇𝑖𝑘, respectively, we have inequality (3.1).
From conditions (3.1) and (3.2), 𝐷+𝑉(𝑥(𝑡))<0. It means that the impulsive switched system is robustly asymptotically stable, except possibly at the impulsive switching points.
Next, for the impulsive switching time point 𝑡𝑘, we have that 𝑉𝑥𝑡+𝑘𝑥𝑡𝑉𝑘=𝑥𝑇𝑡+𝑘𝑃𝑖𝑘𝑥𝑡+𝑘𝑥𝑇𝑡𝑘𝑃𝑖𝑘1𝑥𝑡𝑘=𝑥𝑇𝑡𝑘𝐼+𝐺𝑘𝑇𝑃𝑖𝑘𝐼+𝐺𝑘𝑃𝑖𝑘1𝑥𝑡𝑘.(3.13) Obviously, if (𝐼+𝐺𝑘)𝑇𝑃𝑖𝑘(𝐼+𝐺𝑘)𝑃𝑖𝑘1<0, we have that 𝑉(𝑥(𝑡+𝑘))𝑉(𝑥(𝑡𝑘))<0. On the other hand, by using Schur complement, (𝐼+𝐺𝑘)𝑇𝑃𝑖𝑘(𝐼+𝐺𝑘)𝑃𝑖𝑘1<0 is equivalent to condition (c) of the theorem given by (3.3). Thus, by (3.1), (3.2), and (3.3), the impulsive switched system (2.1) is robustly asymptotically stable.
This completes the proof.

4. Design of Feedback Controller

In this section, we focus on designing a memoryless state feedback controller in the form of 𝑢(𝑡)=𝐾𝑖𝑘𝑥(𝑡), which stabilizes the uncertain impulsive switched systems with discrete and distributed delays considered.

Theorem 4.1. Suppose that there exist symmetric positive definite matrices 𝑃𝑖𝑘, 𝑄𝑖𝑘, 𝑇𝑖𝑘, and 𝑅𝑖𝑘 and some positive scalars 𝜀1>0,𝜀2, such that for 𝑖𝑘=1,2,,𝑚 LMIs (3.2), (3.3), and the following LMIs hold: 𝑄𝑖𝑘00𝑄𝑖𝑘01𝑇𝑖𝑘01𝑇𝑖𝑘02𝑅𝑖𝑘2𝑅𝑖𝑘0𝑍𝑖𝑘𝑃𝑖𝑘𝑋𝑖𝑘𝐼<0,(4.1) where 𝑋𝑖𝑘𝑋𝑇𝑖𝑘=(𝜀1+𝜀2)𝐸𝑖𝑘𝐸𝑇𝑖𝑘𝐷𝑖𝑘𝐷𝑇𝑖𝑘, 𝑍𝑖𝑘=𝑃𝑖𝑘𝐴𝑖𝑘+𝐴𝑇𝑖𝑘𝑃𝑖𝑘+𝜀11𝐻𝑇𝑖𝑘𝐻𝑖𝑘+𝐼.
Then the trivial solution of the impulsive switched system (2.1) is asymptotically stable. Moreover, 𝑢(𝑡)=𝐾𝑖𝑘𝑥(𝑡),𝐾𝑖𝑘1=2𝐷𝑇𝑖𝑘𝑃𝑖𝑘,(4.2) is a feedback controller which stabilizes the corresponding closed-loop impulsive switched system.

Proof. Substitute 𝑢(𝑡)=𝐾𝑖𝑘𝑥(𝑡) and 𝐾𝑖𝑘=(1/2)𝐷𝑇𝑖𝑘𝑃𝑖𝑘 into the system (2.1). Then the corresponding closed-loop impulsive switched system is of the form 𝐴̇𝑥(𝑡)=𝑖𝑘𝑥(𝑡)+𝐵𝑖𝑘𝑥𝑡1+𝐶𝑖𝑘𝑡𝑡2𝑥(𝑠)d𝑠,𝑡𝑡𝑘,Δ𝑥(𝑡)=𝐺𝑘𝑥(𝑡),𝑡=𝑡𝑘,𝑥(𝑡)=𝜑(𝑡),𝑡0,(4.3) where 𝐴𝑖𝑘=𝐴𝑖𝑘(1/2)𝐷𝑖𝑘𝐷𝑇𝑖𝑘𝑃𝑖𝑘+Δ𝐴𝑖𝑘(𝑡).
Replacing 𝐴𝑖𝑘 with 𝐴𝑖𝑘(1/2)𝐷𝑖𝑘𝐷𝑇𝑖𝑘𝑃𝑖𝑘 in the matrix Φ𝑖𝑘 defined in Theorem 3.1, we have that Φ𝑖𝑘=Φ11Φ120𝑃𝑖𝑘𝐶𝑖𝑘𝑄𝑖𝑘1001𝑇𝑖𝑘0102𝑅𝑖𝑘,(4.4) where Φ11=𝑃𝑖𝑘(𝐴𝑖𝑘(1/2)𝐷𝑖𝑘𝐷𝑇𝑖𝑘𝑃𝑖𝑘)+(𝐴𝑖𝑘(1/2)𝐷𝑖𝑘𝐷𝑇𝑖𝑘𝑃𝑖𝑘)𝑇𝑃𝑖𝑘+𝑄𝑖𝑘+1𝑇𝑖𝑘+2𝑅𝑖𝑘+𝑃𝑖𝑘𝐸𝑖𝑘𝐹𝑖𝑘𝐻𝑖𝑘+𝐻𝑇𝑖𝑘𝐹𝑇𝑖𝑘𝐸𝑇𝑖𝑘𝑃𝑖𝑘, and Φ𝑖𝑘𝐼𝑃𝑖𝑘𝐵𝑖𝑘0𝑃𝑖𝑘𝐶𝑖𝑘𝑄𝑖𝑘+𝜀21𝐽𝑇𝑖𝑘𝐽𝑖𝑘1001𝑇𝑖𝑘0102𝑅𝑖𝑘+𝑍𝑖𝑘000000000,(4.5) where 𝑍𝑖𝑘=𝑃𝑖𝑘𝐴𝑖𝑘+𝐴𝑇𝑖𝑘𝑃𝑖𝑘+𝑄𝑖𝑘+1𝑇𝑖𝑘+2𝑅𝑖𝑘+𝜀11𝐻𝑇𝑖𝑘𝐻𝑖𝑘+𝐼+𝑃𝑖𝑘((𝜀1+𝜀2)𝐸𝑖𝑘𝐸𝑇𝑖𝑘𝐷𝑖𝑘𝐷𝑇𝑖𝑘)𝑃𝑖𝑘.
Similar to the proof of Theorem 3.1, in what follows, we will prove that the following LMIs hold: 𝑍diag𝐼,𝐼,𝐼,𝑖𝑘,𝐼<0.(4.6)
Define 𝑊𝑖𝑘=𝑄𝑖1/2𝑘0000011/2𝑇𝑖1/2𝑘0000021/2𝑅𝑖1/2𝑘00𝑄𝑖1/2𝑘11/2𝑇𝑖1/2𝑘21/2𝑅𝑖1/2𝑘𝐼𝑃𝑖𝑘𝑋𝑖𝑘0000𝐼,(4.7) where 𝑋𝑖𝑘 is defined in Theorem 4.1. Then by left multiplying and right multiplying by 𝑊𝑖𝑘 and 𝑊𝑇𝑖𝑘, respectively, we have the LMIs (4.1). The rest of the proof is similar to that of Theorem 3.1 and will be omitted.
This completes the proof.

5. Numerical Example

As an illustration, we consider a system in the form of (2.1) under the given switching rule Δ𝑡𝑘1, 𝑘. Without loss of generality, assume that there are two subsystems, that is, 𝑖𝑘{1,2}, between which the dynamical system alternates. Choose the discrete time delay 1=0.1. We consider robust performance of the system using Theorem 3.1. The parameters of the system are specified as follows: 𝐴1=2002.4,𝐵1=0.200.71,𝐶1=0.500.40.1,𝐷1=,𝐴0.05100.052=2.5001.2,𝐵2=0.7011,𝐶2=0.30.10.60.4,𝐷2=,𝐸0.05100.051=0.1000.1,𝐸2=0.1000.1,𝐻1=0.1000.1,𝐻2=,𝐽0.1000.31=0.2000.1,𝐽2=0.2000.2,𝐺𝑘=.0.1000.1(5.1) Let 𝜀1=𝜀2=1. Then by solving (3.1)~(3.3) under MATLAB Toolbox, we obtain the upper bound of 2=0.1546.

Letting 1=0.1 and 2=0.1546, we obtain the following linear memoryless controller by using Theorem 4.1: 𝐾1=0.11580.01270.01270.0322,𝐾2=0.12160.01820.01820.0315.(5.2)

6. Conclusion

In this paper, the asymptotic stability problem for a class of uncertain impulsive switched systems with discrete and distributed delays is discussed. Firstly, delay-dependent stability criteria have been obtained by choosing proper Lyapunov function. Furthermore, some appropriate feedback controllers have been constructed to ensure the asymptotic stability of the closed-loop systems. A numerical example is solved by MATLAB Toolbox to illustrate that the results obtained are effective.

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